Melt butter over medium heat in a 3- to 4-quart Dutch oven or heavy bottomed saucepan. When it foams, add fennel, onions, and carrots, and cook until just soft and onions are translucent, about 2 minutes. Add mushrooms and potato, season well with salt and freshly ground black pepper, and stir to coat. Cook, stirring rarely, until mushrooms have let off water and are shrunken, about 6 minutes.

Sprinkle flour over vegetables, stir to coat, and cook until raw flavor is gone, about 1 to 2 minutes. Carefully add broth and milk, stirring constantly until mixture is smooth. Bring to a simmer over medium heat and cook until slightly thickened, about 5 minutes.

Remove from heat, add peas, herbs, and vinegar, and stir to coat. Season well with salt and freshly ground black pepper. Turn filling into an 8 by 8-inch baking dish.

Whisk egg together with 2 teaspoons water and a pinch of salt until evenly mixed. Set aside.

With kitchen shears, cut dough to fit over the baking dish. Place dough over filling and tuck into the edges of the dish. Brush dough with egg wash and cut slits in the top to vent. Place on a baking sheet and bake until crust is golden brown and mixture is bubbling, about 25 to 30 minutes. Let sit at least 5 minutes before serving.